====== Gestation 1 Gender ======

\( \require{newcommand} \)
\( \newcommand{\diff}[2]{\frac{d{#1}}{d{#2}}} \)
\( \newcommand{\abs}[1]{\left|#1\right|} \)

Analysis and numerics for case of gestation.  Idealized to a single gender for convenience but (as was found for [[..:2009:Estrous 2 Genders | Estrous]]) we expect the results to generalize to two genders.  We find here that gestation simplifies population dynamics in sexual populations to the logistic rate equation (for large populations, with a weak Allee effect for small populations).  

===== Processes =====

Let \(A\) represent a nonpregnant adult and \(G\) indicate a gestating individual.

|  \( 2 A \xrightarrow{\mu} A + G \)  | impregnation  |
|  \( G \xrightarrow{\beta} 2 A \)  | birth  |
|  \( A + A \xrightarrow{\delta} A \)  | competition  |
|  \( A + G \xrightarrow{\delta} G \)  | competition  |
|  \( G + A \xrightarrow{\delta} A \)  | competition  |
|  \( G + G \xrightarrow{\delta} G \)  | competition  |

The first is a physiological process (impregnation) while the remainder are ecological (affecting the population density).  For the purposes of the analysis, an offspring that is dependent on its parent can also be neglected.  "Birth" in this context means the separation of the child from the parent--when it no longer depends on the parent.

===== Full Dynamics =====

\[
  \diff{a}{t} = - \mu a^2 + 2 \beta g -\delta a n 
\]
\[
  \diff{g}{t} = + \mu a^2 - \beta g - \delta g n 
\]
\[
  \diff{n}{t} = + \beta g - \delta n^2 
\]

where \(n=a+g\).

===== Slow Timescale =====

==== Units ====

Let's assume that ecological processes are much slower than physiological ones.  In particular, let's assume

\[
  \beta, \delta \abs{K} \ll \mu \abs{K} 
\]

<div right 50% round tip>Does it still make sense in this model?  Now gestation is irreversable so there's a one-to-one correspondence between impregnation and birth events.  Need to think about this. </div>

where \( \abs{K} \) is a population density constant, ie. it has the same units as \(n\), \( [K] = [n] \) (where \( [\cdot] \) means "units of").  We choose \( \abs{K} \) so it is a characteristic population size for the dynamics.  That way we can compare the rate constants, because \( K \) matches the units: 

\[
  [\beta ] = [\delta K] = [\mu K]. 
\]

==== Separation of Timescales ====

We can now specify what we mean when we say "ecological processes are much slower than physiological ones".  Let us define \( \hat{\beta} \) and \( \hat{\delta} \) so that

\[
  \beta \equiv \epsilon \hat{\beta}, 
\]
\[
  \delta \equiv \epsilon \hat{\delta} / \abs{K} 
\]

where \(\epsilon \ll 1 \) is a dimensionless constant.  Since \(\epsilon \ll 1\) the ecological processes (involving \(\beta\) and \(\delta\)) occur much less frequently than the physiological transition (with \(\mu\)).

Now we can look at the dynamics under our original (fast) time variable \(t\) or with a new, slow, time variable, \( \hat{t} \equiv \epsilon t \).  In this slower timescale we find the dynamics become

\[
  \epsilon a' \equiv \epsilon \diff{a}{\hat{t}} = -\mu a^2+ 2 \epsilon \hat{\beta} g - \epsilon \frac{\hat{\delta}}{\abs{K}} a n 
\]
\[
  \epsilon g' \equiv \epsilon \diff{ g }{\hat{t}} = + \mu a^2 - \epsilon \hat{\beta} g - \epsilon \frac{\hat{\delta}}{\abs{K}} g n 
\]
\[
  n'          \equiv \diff{n}{\hat{t}} = \hat{\beta} g - \frac{\hat{\delta}}{\abs{K}} n^2. 
\]

where the prime, \( \cdot' \), denotes the slow derivative, \( \diff{\cdot}{\hat{t}} \).

Note that \(\epsilon\) cancels out in the dynamical equation for \(n\) which means that \(n\) varies at a measurable rate on the slow timescale--\(n\) is a //slow// variable.

===== Quasi-Steady-State Assumption =====

So far our analysis has been exact.  Now let's assume that the other, //fast// variables equilibrate quickly so that \( \epsilon a' \approx 0 \approx \epsilon g' \), the quasi-steady-state assumption, QSSA.  Then we can compute the quasi-steady-state densities \(a\) and \(g\) at any time.  Written in terms of the fast parameters (\(\beta\) and \(\delta\)) they are

<div right 50% round tip>The derivation can be found in ''gestation1g20100115.nb''.</div>

\[
  a \approx \frac{-2 \beta -n \delta +\sqrt{(2 \beta +n \delta )^2+8 n \beta  \mu }}{2 \mu } 
\]
\[
  g \approx \frac{\beta +n \delta +2 n \mu -\sqrt{\beta +n \delta } \sqrt{\beta +n \delta +4 n \mu }}{2 \mu }.
\]

Fortunately, this gives us a way to check the validity of our assumption: we originally required \(a+g=n\) so this should still be largely correct.  Approximating as a Taylor series in \(n\) gives the first-order correction:

\[
  a+g \approx n \left[ 1 + \frac{\mu - \delta}{2 \beta } n + \ldots \right] \equiv n \left[ 1 \pm n/n_\epsilon + \ldots \right]. 
\]

This tells us the QSSA may only be valid when

\[
  n \ll n_\epsilon = \frac{2 \beta}{\abs{\delta -\mu}}. 
\]

===== QSSA Birthrate =====

When the QSSA holds we can approximate the population dynamics on the fast, behavioural timescale by

\[
  \diff{n}{t} \approx B(n) n - \delta n^2 
\]

where the per-capita birthrate, \(B(n)\), is
\[
  B(n) \equiv \frac{\beta  \left(\beta +n (\delta +2 \mu )-\sqrt{\beta +n \delta } \sqrt{\beta +n \delta +4 n \mu }\right)}{2 n \mu }. 
\]

\(B(n)\) is a monotonically increasing function that climbs from \(B(0)=0\) to 
\[
  r \equiv B(\infty) = \frac{\beta  \left(\delta +2 \mu -\sqrt{\delta  (\delta +4 \mu )}\right)}{2 \mu }. 
\]

<div right round box>
 {{gestation1gSpecificBirthrate20100115.jpg?600}}

Figure 1: Per-capita birthrate, \(B(n)\) for population dynamics of \(n\) given QSSA assumption.  The function looks qualitatively similar to the estrous birthrate curve.  (Notice that \(\beta > r\) always.)
</div>

We define the Allee domain to be \(n<n_\mathrm{Allee}\) where \(B(n_\mathrm{Allee})=r/2\) which gives a solution

\[
  n_\mathrm{Allee} = \frac{\beta  \left(\delta +2 \mu -3 \sqrt{\delta  (\delta +4 \mu )}\right)}{-2 \delta ^2-8 \delta  \mu +\mu ^2}. 
\]

==== Logistic domain ====

If \(n_\mathrm{Allee} \ll n \ll n_\epsilon\) then the dynamics lie within the //logistic domain// where the per-capita birthrate is roughly constant, \(B(n)\approx r\), and the population dynamics follow

\[
  \diff{n}{t} \approx r n - \delta n^2, 
\]

the logistic rate equation.

==== Allee domain ====

On the other hand, if the population is small, \( n \ll n_\mathrm{Allee} \), then the birthrate is suppressed, the Allee effect.
! Carrying capacity

Regardless of whether QSSA holds we can find the non-trivial equilibrium population density. Since this represents the //characteristic// population size we can choose this to represent our parameter \(K\) from above:

\[
  K \equiv \frac{\beta  \left(\delta +2 \mu -\sqrt{\delta  (\delta +8 \mu )}\right)}{2 \delta  \mu }. 
\]

I haven't done the Jacobian stability analysis but it looks pretty clear that \(n=K\) is only stable when \(K>0\), otherwise only \(n=0\) is stable.  Then if \(K\leq 0\) the population is unstable and will inevitably go extinct.  So we only expect \(K>0\) to represent naturally-occurring populations (unless the parameters have changed recently to destabilize the population).

If \(K>0\) then the //weak// Allee effect applies in the Allee domain so the population is globally stable (regardless of how small \(n\) gets).  On the other hand, if \(K<0\) then the system is unstable for all \(n\).

I haven't been able to find an analytic form for the population dynamics in terms of the \(r\), \(K\), and \(n_\mathrm{Allee}\).  It would be useful for comparison with the logistic equation.

===== Parameter space =====

The possible dynamics can be expressed more concisely by reducing the parameter space to one dimension: 

\[
  x=\delta/\mu. 
\]

We will find there are just four important regions in this parameter space, depending on the relationships between the constants \(n_\epsilon, K\), and \(n_\mathrm{Allee}\).  In terms of \(x\) these constants can be written as

\[
  n_\epsilon = \frac{\beta}{\mu} \frac{2}{\abs{1-x}} 
\]
\[
  K = \frac{\beta}{\mu} \frac{ 2+x-\sqrt{x(8+x)} }{ 2x } 
\]
\[
  n_\mathrm{Allee} = \frac{\beta}{\mu} \frac{ 2+x-3\sqrt{x(4+x)} }{ 1-8x-2x^2 }. 
\]

==== Stability ====

If \(K\leq 0\) then we expect the population to be unstable and to crash (under both the full dynamics and the QSSA approximation).  In terms of \(x\) this is

|  Unstable :  | \( x\geq 1. \)  |

==== QSSA limit ====

If \( K > 0\) (\(x<1\)) then \(K\) represents a characteristic population density (actually, the equilibrium) so it is reasonable to expect the actual density to be on the same order, \( n \sim K \).  Then we can express the QSSA condition as \( K \ll n_\epsilon \).  This condition implies

|  QSSA :  | \( x \gtrsim 0.190. \)  |

==== Logistic vs Allee domain ====

If \(K>0\) then the dynamics will carry \(n\) towards \(K\).  If \(K<n_\mathrm{Allee}\) then the population will always be in the Allee domain with a depressed population growth rate.  This condition is simply

|  Allee:  | \( x \gtrsim 0.431. \)  |

==== Summary ====

<div right round box>
{{:research:draft:2010:gestation1genderparameterspace.png?300}}

Figure 2: Parameter space can be divided into four regions characterizing the dynamics.  The red dots indicates points where the dynamics are computed numerically (see below).
</div>

In conclusion, there are four regions of interest along the \(x\)-axis:

|  unstable :  | \( n_\epsilon \gtrsim n_\mathrm{Allee} > 0 > K. \)  |
|  stable Allee QSSA :  | \( n_\epsilon \gtrsim n_\mathrm{Allee} \gtrsim K > 0 \)  |
|  stable logistic QSSA :  | \( n_\epsilon \gtrsim K \gtrsim n_\mathrm{Allee} > 0 \)  |
|  stable non-QSSA :  | \( K \gtrsim n_\epsilon \gtrsim n_\mathrm{Allee} > 0 \)  |

===== Numerics =====

To test our analytic results we can construct a numeric simulation.  The above analysis suggests the regions we should check numerically (marked in red on Figure 2).  The results, shown below, suggest the QSSA is a more robust approximation than would be expected from the condition \(n\ll n_\epsilon\).  

| Region                                                                   |                                                 |
| Unstable \\ \( n_\epsilon > n_\mathrm{Allee} > 0 > K \) \ \( x = 2.0 \)  |     {{gestation1g20100122_unstable_x2.png?300}} |
| Allee \\ \( n_\epsilon > n_\mathrm{Allee} > K > 0 \) \ \( x = 0.7 \)     |      {{gestation1g20100122_allee_x0.7.png?300}} |
| Logistic \\ \( n_\epsilon > K > n_\mathrm{Allee} > 0 \) \ \( x = 0.3 \)  |   {{gestation1g20100122_logistic_x0.3.png?300}} |
| Non-QSSA \\ \( K > n_\epsilon > n_\mathrm{Allee} > 0 \) \ \( x = 0.1 \)  |  {{gestation1g20100122_stablenon_x0.1.png?300}} |

